We prove that the class of subgroups of Diff^r(S^1) is not closed under taking free products for each 2<= r <=infinity. More specifically, if G is finitely generated non-virtually-abelian group, then (G x Z) * Z does not embed into Diff^r(S^1). We then complete the classification of RAAGs embeddable in Diff^r(S^1), answering a question of Kharlamov (in a paper of M. Kapovich). This is a joint work with Thomas Koberda.

The non-cuspidal points of X_1(N) correspond to isomorphism classes of pairs (E,P), where E is an elliptic curve and P is a point on E of order N. If E has complex multiplication by an order in an imaginary quadratic field K, we say (E,P) is a K-CM point. In this talk, I will give a compete classification of the degrees of K-CM points on X_1(N)_{/K}, where K is any imaginary quadratic field. This is joint work with Pete L. Clark.

Over the last 25 years we have seen incredible advances in the performance of end-user language technologies such as speech recognition and machine translation. However, almost all of the research and engineering effort to date has been expended on 100 or so languages, primarily those of greatest commercial interest: English, French, Chinese, Japanese, German, etc. We'll begin by explaining some of the mathematical models underlying this work, focusing on language modeling as a relatively simple but foundational technology. We'll move on to discuss some of the ways existing models fall short for non-Indo-European languages, the difficulties faced by small language groups in terms of resource-building, and our efforts to overcome some of these difficulties for the next thousand (or more) languages.

Let p and q are relatively prime positive integers with p less than q. A Ford circle C(p/q) is a circle lying in the upper half plane tangent to the point p/q on the real line with radius 1/2q^2. We will show you some interesting features of Ford circles. They never intersect each other. The sum of the area of Ford circles is computable and equal to (pi)(zeta(3))/4zeta(4). If time permits then we will describe the method to count the Ford circles up to a given radius.

Bowditch described the boundary of a relatively hyperbolic group pair $(G,P)$ as the boundary of any hyperbolic space that $G$ acts geometrically finitely upon, where the maximal parabolic subgroups are conjugates of the subgroups in $P$. For example, the fundamental group of a hyperbolic knot complement acts geometrically finitely on $\mathbb{H}^3$, where the maximal parabolic subgroups are the conjugates of $\mathbb{Z} \oplus \mathbb{Z}$. Here the Bowditch boundary is $S^2$. We show that torsion-free relatively hyperbolic groups whose Bowditch boundaries are $S^2$ are relative $PD(3)$ groups. This is joint work with Bena Tshishiku. If time permits, I'll show some examples of strange phenomena that can happen with boundaries of relatively hyperbolic groups, joint with Chris Hruska.

In 1977, Mazur showed that the "Eisenstein quotient" of J_0(p) has rank 0 (and so finitely many rational points). We show that for many primes p, there is a further quotient of J_0(p) such that a positive proportion of quadratic twists also have rank 0. This is a special case of a general result concerning abelian varieties with real multiplication. The proof uses recent our work with Manjul Bhargava, Zev Klagbsrun, and Robert Lemke Oliver, on the average size of the Selmer group of a 3-isogeny in any quadratic twist family.

Algebraic K-theory is a fundamental invariant encoding information about number theory, manifold geometry, and algebraic geometry. However, it is hard to compute directly. Instead, a very successful approach to studying algebraic K-theory has been via "trace methods", which map out to more tractable theories such as (topological) Hochschild and cyclic homology. These theories are interesting in their own right. In this talk, I will give a gentle overview of this story.

I will discuss the geometry and topology of complex hyperbolic 2-manifolds, highlighting open questions and recent progress directly inspired by the last 40 years of work on hyperbolic 2- and 3-manifolds. Emphasis will be on explicit topological constructions (particularly of minimal volume manifolds), fibrations, and betti numbers. Much of this will cover joint work with Luca Di Cerbo.

Arithmetic topology has its origins in an analogy between rings of integers and three dimensional manifolds which goes back to Barry Mazur in the 1960's. In this talk I will discuss some recent developments in the subject. One has to do with an arithmetic version of Chern Simons theory suggested by Minhyong Kim. Another has to do with the application of Massey triple products to Iwasawa theory. These analogies are a two way street. Results in number theory or about three manifolds suggest new questions about the other of the two subjects.

4:00 pm Tuesday, November 7, 2017AGNT: Reduction of dynatomic curves: The good, the bad, and the irreducible
by Andrew Obus (University of Virginia) in HBH 227

The dynatomic modular curves parameterize one-parameter families of dynamical systems on P^1 along with periodic points (or orbits). These are analogous to the standard modular curves parameterizing elliptic curves with torsion points (or subgroups). For the family x^2 + c of quadratic dynamical systems, the corresponding modular curves are smooth in characteristic zero. We give several results about when these curves have good/bad reduction to characteristic p, as well as when the reduction is irreducible. We will also explain some motivation from the uniform boundedness conjecture in arithmetic dynamics.

In this talk I will discuss continuity of eigenvalues and eigenfunctions of self-adjoint Schr\”odinger operators on metric graphs with respect to edge lengths. The standard results in this direction address only the case of strictly positive edge lengths. I will show that most of these results can be carried over to the case of zero limiting lengths.

4:00 pm Wednesday, November 8, 2017Colloquium: Complexity of triviality in topology with applications to geometric variational problems
by Alexander Nabutovsky (University of Toronto) in HBH 227

The first theme of the talk is that some geometric objects have trivial topology, but it is very difficult to see that this, indeed, is the case. The second theme is that this phenomenon implies the ruggedness of some moduli spaces in differential and combinatorial geometry, and that it even forces the existence of non-trivial solutions of some problems in geometric calculus of variations. These phenomena were previously known for dimensions >4 (joint work with Shmuel Weinberger). However, recently we managed to prove that they already exist in dimension 4 (joint work with Boris Lishak). In particular, I will show that (1) it is very difficult to untie some trivial 2-knots in the four-dimensional space; (2) there exist ``many" contractible 2-dimensional complexes, which are extremely difficult to contract and for each pair of them it is also very difficult to see that they are homotopy equivalent to each other; (3) for each n>3 there exist infinitely many distinct local minima of curvature-pinching sup |K| diam^2 (C^0 norm of the sectional curvature normalized by the square of the diameter) on the space of Riemannian structures on the n-sphere; (4) in high-dimensions there exist many non-trivial ``thick" knots of codimension one (unlike the usual knot theory).

Thirty five years ago M. Gromov asked if it is true that the length of a shortest periodic geodesic on a closed Riemannian manifold does not exceed c(n)Vol^{1/n}, where Vol denotes the volume of the manifold, and c(n) is a constant that depends only on its dimension n. This question and a similar question with the diameter of the manifold instead of Vol^{1/n} are still open. I will discuss the solutions to these and related questions in dimension 2, as well as the upper bounds for periodic geodesics, stationary geodesic nets, loops and minimal surfaces in higher dimensions.

4:00 pm Friday, November 10, 2017Undergraduate Colloquium: What I Did Last Summer
by Ilya Marchenka '19 and Anh Tran '19 (Rice University) in HBH 227

Come hear two Rice math majors tell you about their research experiences last summer! They will tell you about their projects and what they enjoyed about the experience. There will also be lots of time for questions so that you can learn about the opportunities available for you! Ilya implemented a prototype of a multiplicatively homomorphic, pairing-based cryptosystem to be used in elections. Anh studied pattern-avoiding permutations using algebraic tools such as characteristic polynomials of linear recurrences.

The tautological ring of the moduli space of curves is a subring of the Chow ring that, on the one hand, contains many of the classes represented by "geometrically defined" cycles (i.e. loci of curves that satisfy certain geometric properties), on the other has a reasonably manageable structure. By this I mean that we can explicitly describe a set of additive generators, which are indexed by suitably decorated graphs. The study of the tautological ring was initiated by Mumford in the '80s and has been intensely studied by several groups of people. Just a couple years ago, Pandharipande reiterated that we are making progress in a much needed development of a "calculus on the tautological ring", i.e. a way to effectively compute and compare expressions in the tautological ring. An example of such a "calculus" consists in describing formulas for geometrically described classes (e.g. the hyperelliptic locus) via meaningful formulas in terms of the combinatorial generators of the tautological ring. In this talk I will explain in what sense "graph formulas" give a good example of what the adjective "meaningful" meant in the previous sentence, and present a few examples of graph formulas. The original work presented is in collaboration with Nicola Tarasca and Vance Blankers.

In this talk I will discuss continuity of eigenvalues and eigenfunctions of self-adjoint Schr\”odinger operators on metric graphs with respect to edge lengths. The standard results in this direction address only the case of strictly positive edge lengths. I will show that most of these results can be carried over to the case of zero limiting lengths.

I will explain a new proof of the non-linear stability of the Minkowski spacetime as a solution of the Einstein vacuum equation. The proof relies on an iteration scheme at each step of which one solves a linear wave-type equation globally. The analysis takes place on a suitable compactification of R^4 to a manifold with corners whose boundary hypersurfaces correspond to spacelike, null, and timelike infinity; I will describe how the asymptotic behavior of the metric can be deduced from the structure of simple model operators at these boundaries. This talk is based on joint work with András Vasy.

4:00 pm Wednesday, November 15, 2017Geometry-Analysis Seminar: Stability results of generalized Beltrami fields and applications to vortex structures in the Euler equations
by David Poyato (University of Granada) in HBH 227

Strong Beltrami fields, that is, 3D velocity fields whose vorticity is the product of itself by a constant factor, are particular solutions to the Euler equations that have long played a key role in Fluid Mechanics. Its importance relies on the Lagrangian theory of turbulence as they were expected to exhibit chaotic configurations. Very recently, this ancient conjecture of Lord Kelvin was positively answered by Alberto Enciso and Daniel Peralta-Salas (ICMAT, Spain). Specifically, there are strong Beltrami fields exhibiting any type of linked vortex lines and tubes of arbitrarily complicated topology. Nevertheless, such existence result is quite tight in the sense that Beltrami fields with non-constant factor (generalized Beltrami fields) are “rare”. Thus, the existence of turbulent configurations is limited to Beltrami fields with a constant factor. The aim of this talk is twofold. First, we will review the state of the art in this topic. Second, we will show that although a full stability result is not possible, there are certain privileged ways to perturb a strong Beltrami field and obtain Beltrami fields with a non-constant factor that even realize arbitrarily complicated vortex structures. This partial stability will be captured in terms of an "almost global” and a “local" stability theorem. The proof relies on analyzing the well-posedness and propagation of compactness and regularity of an innovative iterative scheme of Grad-Rubin type inspired by some numerical methods coming from Astrophysics.

The Brauer dimension of a field F is defined to be the least number n such that index(A) divides period(A)^n for every central simple algebra A defined over any finite extension of F. One can analogously define the Brauer-p-dimension of F for p, a prime, by restricting to algebras with period, a power of p. The 'period-index' questions revolve around bounding the Brauer (p) dimensions of arbitrary fields. In this talk, we look at the period-index question over complete discretely valued fields in the so-called 'bad characteristic' case. More specifically, let K be a complete discretely valued field of characteristic 0 with residue field k of characteristic p > 0 and p-rank n (= [k:k^p]). It was shown by Parimala and Suresh that the Brauer p-dimension of K lies between n/2 and 2n. We will investigate the Brauer p-dimension of K when n is small and find better bounds. For a general n, we will also construct a family of examples to show that the optimal upper bound for the Brauer-p-dimension of such fields cannot be less than n+1. These examples embolden us to conjecture that the Brauer p-dimension of K lies between n and n+1. The proof involves working with Kato's filtrations and bounding the symbol length of the second Milnor K group modulo p in a concrete manner, which further relies on the machinery of differentials in characteristic p as developed by Cartier. This is joint work with Bastian Haase.

The Polya-Vinogradov inequality, an upper bound on character sums proved a century ago, is essentially optimal. Unfortunately, it's also not so useful in applications, since it's nontrivial only on long sums (while in practice one usually needs estimates on sums which are as short as possible). The best tool we have to handle shorter sums is the Burgess bound, discovered in 1957; this is generally considered to supersede Polya-Vinogradov, both because its proof is "deeper" (building on results from algebraic geometry) and because it is more applicable. In this talk I will introduce and motivate both of these bounds, and then describe the unexpected result (joint with Elijah Fromm, Williams '17) that even a tiny improvement of the (allegedly weaker) Polya-Vinogradov inequality would imply a major improvement of the (supposedly superior) Burgess bound. I'll also discuss a related connection between improving Polya-Vinogradov and the classical problem of bounding the least quadratic nonresidue (joint with Jonathan Bober, University of Bristol).

Students from two projects of the Rice Geometry Lab will be doing a poster presentation to showcase what they have done this semester. Prof. Roy and Prof. Li have been directing teams of undergraduate students this semester on the projects 'Music and Geometry’ and 'Understanding the works by Nash on isometric embeddings'. Graduate students, Yikai Chen and Xian Dai, have been helping the teams out. Come take a look at what some of your fellow undergraduate students are doing! We will be looking for interested undergraduate students and people who are interested in directing projects for the participating students. All are welcome and encouraged to attend.